Tensor Abelian Geometry of VI-modules

TL;DR

This paper investigates the spectrum of prime Serre ideals in global representations of noetherian families, revealing a topological structure for finitely generated VI-modules that differs from the Balmer spectrum, with potential applications to FI-modules and cyclic p-groups.

Contribution

It establishes the homeomorphism between the spectrum of prime Serre ideals of finitely generated VI-modules and the one-point compactification of N, providing new insights into their topological structure.

Findings

Spectrum of prime Serre ideals of finitely generated VI-modules is homeomorphic to N^{*}

The method applies to FI-modules and cyclic p-groups

The spectrum differs from the Balmer spectrum of derived VI-modules

Abstract

In this short note, we study the spectrum of prime Serre ideals of global representations for noetherian families. In particular, we prove that the spectrum of prime Serre ideals of finitely generated VI-modules is homeomorphic to N^{*}, the one-point compactification of N, which differs from the Balmer spectrum of derived VI-modules. Our method could also be applied to the category of finitely generated FI-modules and the category of global representations for the family of cyclic p-groups.